AN UPPER BOUND FOR THE X-RANKS OF POINTS OF ℙn IN POSITIVE CHARACTERISTIC

Abstract

Let $X\subset {\mathbb{P}}^n$ be an integral and non-degenerate $m$-dimensional variety. For any $P\in {\mathbb{P}}^n$ the $X$-rank $r_X(P)$ is the minimal cardinality of $S\subset X$ such that $P\in \langle S\rangle$. Here we study the pairs $(X,P)$ such that $r_X(P)\ge n+2-m$, i.e. $r_X(P)=n+2-m$. These pairs exist only in positive characteristic, with $X$ strange and $P$ a strange point of $X$.